MHD motion tends to produce nearly discontinuous magnetic fields and singular current density. In general, an adaptive numerical method is needed. We have solved two dimensional, incompressible MHD equations on an unstructured grid of triangles, using a a piecewise linear finite element discretization. A stream function representation of the magnetic and velocity fields is used to ensure zero divergence of the fields.
A straightforward application of this approach causes problems with the current, due to the local non convergence of the finite element Laplacian. An improved Laplacian can be constructed by successive application of the finite element gradient and divergence. For adaptive computations, the best results are obtained by reformulating the MHD equations so that the vorticity and current are time advanced, with the magnetic and velocity stream functions found by solving Poisson equations.
For adaptive computations, mesh operations are provided to reconnect triangles and to refine (and unrefine) the mesh. The mesh adaptively refines to resolve current sheets. Example simulations of the coalescence and tilt instability show the formation of current sheets, with the current density increasing exponentially in time.
The unstructured mesh methods described in this paper have also been applied to problems in which the computational boundary has a complicated shape. The methods are being incorporated in a three dimensional MHD code, in which the mesh is unstructured in two dimensions, and structured in the third dimension.
Acknowledgment
We wish to thank Anne Greenbaum of NYU, who provided us with a sparse matrix ICCG routine. This work was supported by AFOSR Grant No. 91-0044 and U. S. Department of Energy Grant DE-FG02-86ER53223. Some of the computations were performed at NERSC, operated by the USDOE.